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16 Sep 2014, 00:27
Official Solution:If \(\frac{t}{u} = \frac{x}{y}\) and \(\frac{t}{y} = \frac{u}{x}\) and \(t\), \(u\), \(x\), and \(y\) are nonzero integers, which of the following is true? A. \(\frac{t}{u}=1\) B. \(\frac{y}{x}=1\) C. \(t = u\) D. \(t = \pm u\) E. None of the above Given that: \(\frac{t}{u} = \frac{x}{y}\) and \(\frac{t}{y} = \frac{u}{x}\). So, \(\frac{t}{u} = \frac{x}{y}\) and \(\frac{t}{u} = \frac{y}{x}\) (from 2), which means that \(\frac{t}{u}\) and \(\frac{x}{y}\) equal to their reciprocals: \(\frac{t}{u}=\frac{u}{t}\) and \(\frac{x}{y}=\frac{y}{x}\). So, \(t^2=u^2\) and \(t^2=u^2\), which gives \(t=u\) (or which is the same \(t = \pm u\)) and \(x=y\) (or which is the same \(x = \pm y\)). Answer: D
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Re: M0616
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13 Feb 2015, 22:18
Eqn 1....Let t/u = x/y = k
Therefore t = uk, x = yk.....
Substitute in eqn 2...t/y = u/x
uk/y = u/yk
k^2 = 1
k = +/1
so t = +/ u



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Re: M0616
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17 Feb 2015, 11:55
Bunuel In questions where they ask which is true.. are they asking for must be true or which can be true? Asking since i solved this question as which can be true and stopped after choosing A as A was coming as true for a case where all numbers were equal.



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10 Jun 2015, 11:54
Why are we crossmultiplying ? The question does not state that integers t,x,u,v are positive...



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11 Dec 2015, 05:31
Hi,
I did the following:
From (1), ty=xu (cross multiplying) From (2), tx=uy
so adding each side of the equalities, t(x+y)=u(x+y) so t=u (and not t= +u)
What am I doing wrong?



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26 Apr 2016, 18:39
m9338 wrote: Hi,
I did the following:
From (1), ty=xu (cross multiplying) From (2), tx=uy
so adding each side of the equalities, t(x+y)=u(x+y) so t=u (and not t= +u)
What am I doing wrong? t=u would be true for negative and positive values of u. But I can see that some people would choose E.



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Re: M0616
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15 Jun 2016, 02:34
m9338 wrote: Hi,
I did the following:
From (1), ty=xu (cross multiplying) From (2), tx=uy
so adding each side of the equalities, t(x+y)=u(x+y) so t=u (and not t= +u)
What am I doing wrong? One explanation that i can offer here is as follows  The equation t(x+y)=u(x+y) holds good in almost all case, however (given x,y are non zero integers), consider a scenario where x= y. In this case you cannot say that t=u as you cant divide both sides with (x+y) !!! why? coz its 0 Incidentally this is one of the wrong answer choices as well
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Re: M0616
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18 Feb 2017, 07:53
I still can't understand why option D is correct, even though the question is MUST BE TRUE. Here is my thinking:
Let's say t, u, x and y are all equal to 1 (nonzero integer). Then according to option D: 1 = +1, which is only true for positive sign, but not negative. I mean the option D would be a MUST BE TRUE answer only if it was written like this: t = u OR t = u. As far as I understand t = +u means 't = u AND t = u'



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10 Jul 2018, 19:42
why is t = u the same thing as t = ±u? I don't understand that. Can someone please explain?



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10 Jul 2018, 20:27



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Re: M0616
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13 Aug 2018, 07:24
m9338 wrote: Hi,
I did the following:
From (1), ty=xu (cross multiplying) From (2), tx=uy
so adding each side of the equalities, t(x+y)=u(x+y) so t=u (and not t= +u)
What am I doing wrong? I still cannot understand why this is wrong. Could someone help me?



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Re: M0616
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05 Sep 2018, 06:40
So, t^2=u^2 and t^2=u^2 gives t=u
Could someone please elaborate the result derived above?



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05 Sep 2018, 06:45
Megha1119 wrote: So, t^2=u^2 and t^2=u^2 gives t=u
Could someone please elaborate the result derived above? By taking the square root. Remember: \(\sqrt{x^2}=x\). The point here is that square root function can not give negative result: wich means that \(\sqrt{some \ expression}\geq{0}\). So \(\sqrt{x^2}\geq{0}\). But what does \(\sqrt{x^2}\) equal to? Let's consider following examples: If \(x=5\) > \(\sqrt{x^2}=\sqrt{25}=5=x=positive\); If \(x=5\) > \(\sqrt{x^2}=\sqrt{25}=5=x=positive\). So we got that: \(\sqrt{x^2}=x\), if \(x\geq{0}\); \(\sqrt{x^2}=x\), if \(x<0\). What function does exactly the same thing? The absolute value function! That is why \(\sqrt{x^2}=x\)
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Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
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